Theoretical insights into the structural, optoelectronic, thermoelectric, and thermodynamic behavior of novel quaternary LiZrCoX (X = Ge, Sn) compounds based on first-principles study

The structural, magnetic, electronic, elastic, vibrational, optical, thermodynamic as well as thermoelectric properties of newly predicted quaternary LiZrCoX (X = Ge, Sn) Heusler compounds are evaluated intricately with the aid of ab initio techniques developed under the framework of density functional theory. The computed structural properties are found to be in tandem with the existing analogous theoretical and experimental facts. Structural optimization has been carried out in three different structural arrangements, i.e., Type-1, Type-2, and Type-3. Further analysis of the optimization curves reveals that the Type-3 phase, which has the least amount of energy, is the most stable structure for the compounds under consideration. The tabulated cohesive energy and formation energy of these compounds depict their chemical as well as thermodynamic stability. The absence of negative phonon frequencies in the phonon band spectrum of the studied compounds depicts their dynamic stability. Similarly, the tabulated second-order elastic constants (Cij) and the linked elastic moduli show their stability in the cubic phase. The calculated value of Pugh's ratio and Cauchy pressure reveal that LiZrCoGe is brittle whereas LiZrCoSn is ductile. Additionally, the optical characteristics of the compounds are studied in terms of the dielectric function, refractive index, extinction coefficient, absorption coefficient, reflectivity, energy loss function, and optical conductivity. The obtained high value of power factor and figure of merit of the studied lithium-based quaternary compounds predict good thermoelectric behavior in these compounds. Thus, LiZrCoX (X = Ge, Sn) compounds can therefore be used to create innovative and intriguing thermoelectric materials as well as optoelectronic and energy-harvesting equipment.


Introduction
3][4] The power factor (S 2 s) or gure of merit (ZT) can be increased by diminishing the value of the thermal conductivity of the solid and increasing electrical conductivity.Low thermal conductivity k l can be accomplished by isoelectronic alloying, defect engineering, etc. [5][6][7] With the advent of science and advances, it is obligatory to design operative thermoelectric alloys with augmented electronic, elastic, and mechanical properties.The discovery of a novel class of Heusler compounds has made it possible to realize such prospects for the advancement of modern technology.Heusler compounds are mainly of three categories, i.e., ternary Heusler alloys (ABC), full Heusler alloys (A 2 BC), and quaternary Heusler alloys EQH (AA ′ BC). 8 When one of the A atoms in the full Heusler alloys is replaced with another type of atom A ′ , EQH alloys are obtained.EQH alloys possess less disorder in comparison to the ternary Heusler. 9The EQH alloys, however, are created by lling any suitable voids between half Heuslers or full Heuslers, which comply with the 18-Valence Electron Count (VEC) rule, using an appropriate electropositive element, such as Li.They can be easily tuned for designing various electronic as well as transport properties for varied applications in optoelectronics, thermoelectric devices, etc.1][12][13][14] First-principles calculations were utilised by Haleoot et al. (2020) to analyse the exceptional thermoelectric and thermodynamic properties of the Quaternary Heusler (QH) compounds CoFeYGe (Y = Ti, Cr). 15 Density functional theory (DFT) has been used to examine another novel quaternary Heusler compound, CrVNbZn, which has the space group 216 (of cubic geometry) and shows maximum ZT = 0.79 in the wide temperature range of 260 K to 480 K as well as excellent electronic and magnetic properties. 16ecently, the structural, electronic, and magnetic properties of the equiatomic quaternary Heusler alloy ZnCdRhMn using the rst-principles calculations and Monte Carlo simulations have been investigated by Idrissi et al. 17 Recently, rst principlesbased calculations have been performed for Li-based quaternary compounds LiHfCoX (Ge, Sn) by Kaur et al. 14 Abraham et al. have conducted a comparative analysis of various exchange-correlation functionals aimed at predicting the structural, electrical, optical, and transport features of the new quaternary LiTiCoSn.18 Gupta et al. have investigated the ground state characteristics of a semiconducting new quaternary Heusler alloy LiScPdPb.19 Singh et al. have investigated the vibrational, thermoelectric transport, and mechanical properties of the lithium-based Heusler compound LiTiCoSn using a plane wave pseudopotential approach.20 Structural, electronic, mechanical, and thermoelectric properties of LiTiCoX (X = Si, Ge) compounds have also been investigated by Singh et al.21 Some of the quaternary Heusler compounds based on Li that have recently undergone productive examination 39,40 for their structural, electrical, and thermoelectric properties are LiScPdPb, LiTiCoSi, and LiTiCoGe.Because of the high melting point as well as complying with the typical 18 valence electron counts, the Li based EQH has the capacity to design highefficiency thermoelectric devices. Inlight of these motives, in this work, we have constrained ourselves to the study of lithiumbased EQH LiZrCoX (X = Ge, Sn) alloys.The comparatively smaller atomic size of the lithium (Li) atom is very apt for lling the vacant space in the Heusler alloy.The single valence electron provided by Li, 4 valence electrons from Zr and Ge/Sn, and nine electrons from Co form a 18 valence electron quaternary Heusler system.As per the literature survey, neither experimental nor theoretical investigations have been conducted on these compounds, which motivated us to initiate studies on them.Using rst-principles calculations along with BoltzTraP and PHONOPY, the structural, electrical, mechanical, optical, vibrational, and transport features of these novel compounds LiZrCoX (X = Ge, Sn) are investigated.Section 2 in the current paper describes the calculation procedure, and Section 3 discusses the ndings. Fially, Section 4 brings everything together with the conclusions.

Computational details
The full potential linear augmented plane wave (FP-LAPW) method 22 and the DFT framework have been used to determine the physical characteristics of LiZrCoX (X = Ge, Sn) quaternary Heuslers.The generalized gradient approximation (GGA) is used to predict the structural, elastic, and electronic properties in the WIEN2k code. 23,24Birch-Murnaghan equation of state is used to determine structural parameters by tting the energy vs. volume curve.The R MT K max value is chosen to be 8, and the number of k-points in the rst Brillouin zone is the 15 × 15 × 15 mesh according to the Monkhorst-Pack scheme. 25The valence wave functions are expanded to l max = 10 partial waves, inside the atomic spheres.Up to G max = 12 a.u −1 , the potential and charge density are expanded.The IRELAST method, 26 implemented in the WIEN2k package, is used to compute the elastic constants.The Seebeck coefficient (S), electrical, and thermal conductivities, as well as other thermoelectric properties of these materials as a function of temperature are also computed using the semiclassical BoltzTraP algorithm. 27The rigid band approximation and constant relaxation time are the fundamental ideas behind this code.We also determine the cubic elastic mechanical stability using the relaxed structure derived using the PBE-GGA approach.We were able to get thermodynamic parameters such as the Debye temperature by solving the Gibbs function within the quasi-harmonic Debye model using the Gibbs2 code. 28The quasi-harmonic Debye model for LiZrCoX (X = Ge, Sn) is used in the Gibbs program to examine the thermal effects over the temperature range of 0 to 1200 K. Additionally, the VASP soware is used to generate the cubic LiZrCoX phonon spectrum (X = Ge, Sn) using the pseudopotential plane-wave method. 29The computation uses a 450 eV plane-wave cutoff.For the computations, an 18 × 18 × 18 k-mesh that is a denser mesh is employed for the IBZ integration of the unit cell.

Structural properties
The investigated quaternary Heusler compounds LiZrCoX (X = Ge, Sn) crystallize in cubic phase with space group F43m (SG#216), as shown in Fig. 1. 30,31 There are three possible arrangements of atomic sites, namely Type-1, Type-2, and Type-3.The Wyckoff positions corresponding to these three arrangements are tabulated in Table 1. 32The volume optimization of these compounds has been performed in these three arrangements, and the variation of total energy is plotted against volume for both the investigated compounds.The energy vs. volume curves of LiZrCoX (X = Ge, Sn) are displayed in Fig. 2(a) and (b).The energy vs. volume plot for LiZrCoGe overlaps in Type-1 and Type-2 phases.It can be deduced from Fig. 2 that both the investigated systems crystallize in Type-3 arrangement as the energy-volume curve lies below the other two arrangements.The optimized lattice parameters along with equilibrium energy and bulk modulus and its pressure derivative of the investigated LiZrCoX (X = Ge, Sn) are presented in Table 2. From the equilibrium energy values presented in Table 2, it can be inferred that the studied compounds have minimum energy in the Type-3 phase. 33ll of our subsequent research has been based on a Type-3 setup, with PBE-GGA serving as the exchange-correlation potential and Birch-Murnaghan equation of state being used to match the optimized energy vs. volume data. 34The obtained data of ground state properties are tabulated in Table 3.The optimized lattice parameters, unit-cell volume, equilibrium energy, and bulk modulus and its pressure derivative along with bond length, formation energy, and cohesive energy are presented in Table 3.The lattice parameter of LiZrCoSn is found to be larger than that of LiZrCoGe, which might be due to the increase in the size of Sn than Ge.The bulk modulus of LiZ-rCoGe is greater than LiZrCoSn using PBE-GGA exchange correlation potential; hence, LiZrCoGe is found to be stiffer than LiZrCoSn.The bond length of both the studied LiZrCoX (X = Ge, Sn) has also been computed and presented in Table 3.The bulk modulus of the investigated LiZrCoX (X = Ge, Sn) decreases as we go from Ge to Sn due to their elongated bond  Table 2 Calculated lattice parameter a (Å), volume (Å 3 ), optimized minimum energy (E min ), bulk modulus (B) GPa, pressure derivative (B ′ ) of XX ′ YZ alloy (where X = Li, X ′ = Ti, Y = Co, and Z = Ge/Sn) lengths.The thermodynamic and chemical stabilities of the explored quaternary Heusler LiZrCoX (X = Ge, Sn) alloys have been evaluated by calculating their cohesive energy and formation energy, which is illustrated in Table 3. Cohesive energy can be used for quantifying the atomic binding stability of compounds.The calculation for the cohesive energy per atom is as follows: where E LiZrCoX is the total energy of the unit cell in the cubic phase and E Li , E Zr , E Co , and E X are the energies of individual atoms.][37] The formation energy per atom is evaluated with the help of the following relation: is the total energy of LiZrCoX (X = Ge, Sn) per formula unit, and E bulk Li , E bulk Zr , E bulk Co , and E bulk X are the total energies of Li, Zr, Co, and Ge/Sn bulk, respectively.It is observed from Table 3 that the investigated compounds have negative formation energy, demonstrating that their thermodynamic stability and formation process involves exothermic phenomenon.The evaluated formation energy per atom and cohesive energy per atom of LiZrCoX (X = Ge, Sn) shown in Table 3 reveal their stability in the cubic phase.
Phonon dispersion (PD) against momentum is computed to verify the stability of the LiZrCoX (X = Ge, Sn), as shown in Fig. 3(a) and (b).The modes in the phonon dispersion plot generally look like spaghetti with transverse and longitudinal modes.We used momentum along the x-axis and frequency (THz) along the y-axis in the range of 0 to 6 THz.We discovered a small amount of negative frequency in the LiZrCoX (X = Ge, Sn) compounds, although almost all modes are positive and have actual phonon branches.According to the literature, Table 3 Calculated lattice parameters (a in Å), unit-cell volume (V 0 , in Å 3 ), equilibrium energy (E min in Ry), bulk modulus (B in GPa) and its pressure derivative B ′ , bond length (Å), formation energy (DH, in kJ mol −1 ) and cohesive energy (E coh , in eV per atom)   LiZrCoX (X = Ge, Sn) will be stable if modest pressure levels (GPa) are applied. 38,39

Magnetic properties
The valence electronic conguration of the investigated Li-Zr quaternary alloys LiZrCoX (X = Ge, Sn) includes Li[2s 1 ], Zr [5s 2 4d 2 ], Co[3d 7 4s 2 ], Ge[4s 2 4p 2 ], Sn[5s 2 5p 2 ], forming 18 valence electron count alloys.1][42] The result of spin polarized computations reveals that the total magnetic moment of both the investigated compounds are found to be zero, which agrees well with the Slater-Pauling rule (M T = Z T − 18) m B , in which M T denotes the total magnetic moment, Z T is the count of total valence electrons in the compound, and m B is Bohr magneton. 43,44Therefore, our further investigation of compounds has been carried out in the non-magnetic phase.

Electronic properties
The optimized lattice parameters obtained using PBE-GGA are used to elucidate the electronic properties using band structure as well as density of states plots for the investigated LiZrCoX (X = Ge, Sn).The electronic band structures and density of states are plotted along the high symmetry principal directions along R-G-X-M-K and presented in Fig. 4  a semiconducting nature.The computed effective masses for LiZrCoX along the high symmetry principal directions of the Brillouin zone are tabulated in Table 3.
Studying the effective masses of carriers (holes and electrons) is crucial for improving the understanding of photovoltaic properties, which are highly inuenced by resistivity, carrier mobility, and optical response of free carriers.In LiZrCoX (X = Ge, Sn), the effective mass of carriers is calculated using the following formula.
The estimated effective mass values for both the investigated LiZrCoX compounds are provided in Table 3.The ndings demonstrate that the examined quaternary half Heusler alloys have very small effective masses of carriers (electron and hole).The decreased effective mass is highly benecial for solar

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RSC Advances materials since it makes carrier transfer easier.Accordingly, LiZrCoX (X = Ge, Sn) may be effective in photovoltaic applications.

Elastic and mechanical properties
Both the investigated LiZrCoX (X = Ge, Sn) crystallize in the cubic phase; therefore, they possess three independent secondorder elastic constants (SOECs), namely, C 11 , C 12 , and C 44 .The calculated SOECs of LiZrCoX using PBE-GGA are tabulated in Table 4.The computed SOECs obey Born-Huang stability criteria 46,47 given by which reveals the mechanical stability of the investigated LiZrCoX (X = Ge, Sn) compounds in the cubic phase under ambient conditions.These elastic constants help us to evaluate the mechanical durability, elasticity, and strength to defend the deformation forces on the studied compounds.C 44 refers to the resistance to stress-oriented deformations and C 11 quanties the stiffness of the materials against applied stress on them. 48The computed shear modulus as well as Young's modulus are higher for LiZrCoSn than that of LiZrCoGe, predicting LiZrCoSn to be stiffer than LiZrCoGe.Table 4 presents the computed values of the other related elastic constants, such as shear modus (G), Young's modulus (Y), Poisson's ratio, and anisotropic factor (A). Zener anisotropy factor A can be computed using A = 2C 44 /(C 11 − C 12 ), and the obtained values for both compounds show deviation from unity, depicting their anisotropic behavior. 49The ionic/covalent and ductile/brittle nature of the studied quaternary LiZrCoX (X = Ge, Sn) Heusler compounds can be assessed using certain criteria like Poisson's ratio, Pugh's ratio, and Cauchy pressure.Ductile compounds generally yield Poisson's ratio s > 0.33. 50The value of s for central forces in solids and ionic crystals can vary between 0.25 and 0.5, 50 whereas for covalent materials, the value of s varies between 0.1 and 0.25, and interatomic forces belong to non-central forces. 51,52It can be deduced from Table 4 that the computed value of s is found to be 0.23 and 0.26, respectively, for LiZrCoGe and LiZrCoSn, revealing that LiZrCoGe belongs to ionic and LiZrCoSn belongs to covalent materials.According to Pugh's criterion, 53,54 the material has a ductile nature if its Pugh's ratio is greater than 1.75 and vice versa.The calculated Pugh's ratio of LiZrCoGe is 1.58 and that of LiZrCoSn is 1.77, as given in Table 4, conrming the brittle nature of LiZrCoGe and the ductile nature of LiZrCoSn.The brittleness of LiZrCoGe is conrmed by the negative value of Cauchy pressure (C 12 − C 44 ), and the ductility of LiZrCoSn is also affirmed by the positive value of Cauchy pressure. 55,56Besides these, we have also computed longitudinal sound velocity, transverse sound velocity, and average sound velocity of LiZrCoX (X = Ge, Sn).Using these sound velocities, we have calculated Debye temperature (q D ), a crucial physical entity, which relates various physical properties such as specic heat capacity, thermal conductivity, and melting point of the crystal with elastic constants.At low temperatures, Debye temperature q D can be evaluated from the SOECs.The Debye temperature q D is found to be lower for LiZrCoGe than LiZrCoSn, revealing the possibility of its low lattice thermal conductivity as the Debye temperature is directly related to lattice thermal conductivity. 57The melting points corresponding to these explored LiZrCoX compounds are also evaluated using T melt = [553 K + (5.911K GPa −1 )C 11 ] ± 300 K and tabulated in Table 4.

Optical properties
The optical characteristics of both the investigated quaternary Heusler compounds LiZrCoX (X = Ge, Sn) are computed and presented in Fig. 6(a)-(h), and the values are tabulated in Table 5.The optical properties are essential for determining the optical performance of the devices, especially for optoelectronics and solar cell applications.9][60] The dielectric function 3(u) depicts how electromagnetic radiation interacts with a given material and how the material responds to it.The value of 3 1 (u) and 3 2 (u) can be evaluated using the following relations: in which V is the unit cell volume, m is the mass of electron, e is its electronic charge, jkn is the crystal wave function, p is the momentum operator, f(kn) is the Fermi-Dirac distribution function, and ħu is the incident photon energy.
The optical functions such as the electron loss function L(u) and n(u) can be determined using Kramers-Kronig relations. 61,62uÞ ¼ The computed photon spectral dependency of the real as well imaginary part of the complex dielectric function 3(u) is consolidated in Fig. 6(a) and (b) for both the investigated compounds LiZrCoX (X = Ge, Sn).The real part 3 1 (u) is deduced from the imaginary part 3 2 (u), and 3 2 (u) represents the absorption behavior of the material. 63The static dielectric  79 12.32 3.51 function 3 1 (0) yields the value of 20.97 and 22.12 for LiZrCoGe and LiZrCoSn, respectively.Other reports are in accordance with the computed values for LiZrCOX (X = Ge, Sn).It gives information regarding the electronic polarizability of the materials.The main peak of 3 1 (u), as in Fig. 4(a) of LiZrCoGe, is 2.0 eV, which gets shied to 1.7 eV for LiZrCoSn.Aer the optimal peak, 3(u) diminishes gradually and reaches 0 for LiZ-rCoGe at 3.0 eV and LiZrCoSn at 2.5 eV.It remains in the negative region up to 12 eV for both the investigated quaternary compounds.The spectral dependency of the imaginary part 3 2 (u) with the incident electromagnetic radiation up to 12 eV for both compounds is displayed in Fig. 4(b).The rst peak of 3 2 (u) of LiZrCoSn occurs at almost 2.0 eV for both compounds whereas the optimal peak occurs at about 3.0 eV for both Li-Zr compounds.3 2 (u) decreases gradually with the increase in energy of incident electromagnetic radiation even though minor peaks occur at 4 eV, 5 eV, and 7 eV.
The spectral dependency of the optical conductivity s(u) of the studied LiZrCoX compounds has been investigated and presented in Fig. 6(c).The optimal peak of optical conductivity occurs at 2.3 eV and 3.0 eV for LiZrCoSn and LiZrCoGe, respectively.Several minor peaks of s(u) occur at various points of the incident photon energy of electromagnetic radiation between 3.0 eV and 12.0 eV. 64he incident photon spectral dependency of the absorption spectrum a(u) has been computed and illustrated in Fig. 6(d).It displays the optimal peak in the high energy region of the electromagnetic spectrum, depicting them as prominent UV absorbing materials. 63he refractive index n(u) proles of the studied compounds are illustrated in Fig. 6(e).The static refractive index value n(0) is 4.60 and 4.70 for LiZrCoGe and LiZrCoSn, respectively, and the same is tabulated in Table 5.Other reports are consistent with the LiZrCoX (X = Ge, Sn) calculated values.It increases gradually with incident electromagnetic energy and reaches an optimal value of almost 6.8 around 2 eV for both the investigated compounds and thereaer decreases with the increase in energy of incident electromagnetic radiations.The refractive index prole follows a similar trend as that of the real part of the dielectric function for both explored materials.
The reectivity spectra R(u) with incident electromagnetic radiations for LiZrCoX have been also investigated and presented in Fig. 6(f), which reveals optimal reection at about 5.8 eV for LiZrCoSn and a shi towards the high energy region, i.e., to 8.3 eV in the case of LiZrCoGe.The extinction coefficient k(u) has also been computed for the studied compounds up to incident energy of 12 eV and demonstrated in Fig. 6(g).It is found to have a threshold value below their band gaps and thereaer gradually increases and reaches an optimal value of almost 4 at about 3 eV for both LiZrCoGe and LiZrCoSn, 65 aer which it continuously decreases in the high energy region of the electromagnetic spectrum and follows a similar trend to that of 3 2 (u).The energy loss function L(u) has also been computed in the incident photon energy ranging from 0 to 12 eV and is presented in Fig. 6(h).It describes the amount of energy being lost while a fast-traveling electron traverses through the compound.The maximum energy loss function L(u) has been found to have a value of 0.3 at 10 eV for LiZrCoGe and it gets shied towards the high energy region for LiZrCoSn.Zero energy loss is observed in the band gap region with a plodding rise of incident photon energy for both these materials.

Thermodynamic properties
The thermal as well as pressure response of materials is also very crucial as they play a vital role in their structural, elastic, mechanical, electronic, and vibrational properties.The thermodynamic variables such as specic heat, entropy, Debye temperature, and thermal expansion coefficient were explored using GIBBS soware, 28 which uses quasi-harmonic Debye approximation, and are presented in Fig. 7(a)-(e) and 8(a)-(e) in the case of LiZrCoGe and LiZrCoSn, respectively, in the range up to 1200 K and 80 GPa.As observed from Fig. 7(a) and 8(a), the volume of compounds is found to be increasing with the applied temperature, revealing that they undergo expansion with the temperature.As the pressure increases, the volume gradually decreases for both compounds.The magnitude of the volume of the unit cell of LiZrCoSn is larger compared to that of LiZrCoGe, which might be due to the increased atomic size of the X atom while going from Ge to Sn. 66 The temperature dependence of the bulk modulus for both compounds is illustrated in Fig. 7(b) and 8(b).As observed in Fig. 7 and 8, the bulk modulus slightly decreases with the temperature whereas a sharp dependence with pressure is observed for both compounds. 67The temperature dependence of the heat capacity of both compounds is computed and presented in Fig. 7(c) and 8(c).It is observed that both compounds follow the Dulong-Petit law at higher temperatures.It is seen that C v increases rapidly with temperatures up to 800 K,

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which can be elucidated using Debye's model C v f T 3 ; beyond 800 K, their behavior follows the Dulong-Pettit law C v = 3R where R is the universal gas constant.This rule states that C v reaches a constant value of 99.5 J K −1 mol −1 and 99.98 J K −1 mol −1 for LiZrCoGe and LiZrCoSn, respectively, aer 800 K.However, when the temperature increases, the entropy (S), which represents the degree of disorders (see Fig. 7(d) and 8(d)) also increases and decreases with the pressure in both the explored compounds. 68he temperature dependence of the Debye temperature q D is also studied for both compounds and illustrated in Fig. 7(e) and 8(e), and it was found to decrease with the temperature but increase with the pressure.The temperature dependence of the thermal expansion coefficient a of both compounds has been studied and (see Fig. 7(f) and 8(f)) observed to have a linear increase up to 200 K and gradual increase up to 1200 K at 0 GPa for LiZrCoGe, whereas for LiZrCoSn, aer 400 K, for all pressures, it becomes almost constant till 1200 K.

Thermoelectric properties
The thermoelectric properties of materials rely on the energy dispersion relations and particularly on the nature of E(K) at the Fermi level.The comportment of thermoelectric materials can be assessed by their gure of merit (ZT).It is used to determine the leeway of using an alloy for thermoelectric applications. 69,70he thermoelectric transport properties such as Seebeck coef-cient, electrical conductivity, thermal conductivity, and power factor of LiZrCoX (X = Ge, Sn) have been studied with the help of semiclassical Boltzmann theory as implemented in BoltzTraP code. 27The temperature and chemical potential dependence of these transport properties of both the investigated compounds have been performed to determine whether they can be assessed as potential candidates for the various thermoelectric applications and illustrated in Fig. 9 and 10.The Seebeck coefficient is one of the thermoelectric parameters that is used to determine the charge carrier responsible for conduction in the thermoelectric transport mechanism.The variation in the Seebeck coefficient with the temperature at the Fermi level has been investigated within the range of 0 to 1200 K and is displayed in Fig. 9(a).It is observed that the value of the Seebeck coefficient at 0 K is 254 mV K −1 and 279 mV K −1 for LiZrCoGe and LiZrCoSn, respectively, which gets reduced to 247 mV K −1 and 251 mV K −1 at 300 K, and the same is tabulated in Table 6 with  71 The value of the Seebeck coefficient is found to be continuously decreasing up to 1200 K as seen in Fig. 9(a) for both materials.
The electrical conductivity with respect to the constant relaxation time, i.e., s/s of these compounds, was also studied up to 1200 K and is presented in Fig. 9(b).The s/s curves are found to be overlapping for both compounds.It is observed that the electrical conductivity of the studied materials is found to be gradually increasing with temperature, which again conrms the semiconducting behavior of these compounds. 72,73The electronic thermal conductivity of the studied compounds is also computed and its dependence on temperature is displayed in Fig. 9(c).The total thermal conductivity k tot is the sum of electronic thermal conductivity k Elec and lattice thermal conductivity k Latt .The electronic contribution to the total thermal conductivity for both compounds have been computed using BoltzTraP code, and the lattice contribution is computed using Slack equation as k The variation in k tot , k Elect , and k Latt with temperature has been studied for both compounds and is displayed in Fig. 9(d).The electronic thermal conductivity for both compounds is found to be linearly increasing with temperatures up to 1200 K, which measures the free electron vibrations in the explored materials, whereas the lattice thermal conductivity decreases sharply with the temperature due to the scattering of phonons. 74The total thermal conductivity k tot for both the LiZrCoX compounds is found to be linearly decreasing up to 400 K, aer which it increases with temperatures up to 1200 K. Low thermal conductivity and high electrical conductivity are characteristics of good thermoelectric materials.Therefore, the studied LiZ-rCoGe and LiZrCoSn compounds are good thermoelectric materials up to 400 K.The variation of power factor (PF) with temperature has been studied for both compounds and is displayed in Fig. 9(d).The power factor curves overlap for both compounds and gradually increase with temperature and found to have 1.9 × 10 12 W K −2 m −1 s −1 and 1.6 × 10 12 W K −2 m −1 s −1 for LiZrCoGe and LiZrCoSn, respectively.The gure of merit (ZT) is one of the crucial metrics to determine whether the material is a good thermoelectric or not.Its dimensionless relation is given as ZT ¼ S 2 s k tot T. The calculated gure of merit is found to gradually increase with the temperature and attains a maximum value of 0.78 and 0.76 for LiZrCoGe and LiZrCoSn, respectively, as presented in Table 6, with comparison to available reports.The chemical potential dependence on the Seebeck coefficient has been studied and illustrated in Fig. 10(a).The doping range of material is represented as positive for n-type and negative for p-type along the chemical potential axis (m-3 F ).It can be seen from Fig. 10(a) and (b) that Seebeck coefficients are found to be decreasing with the temperature for both compounds. 75,76The maximum value of the Seebeck coefficient is found to be 1821 mV K −1 for LiZrCoGe at 300 K and 0.5 eV compared to that of 1580 mV K −1 for LiZ-rCoSn.8][79] The electrical conductivity per relaxation time (s/s) of both compounds are studied in the chemical potential ranging from −2.0 eV to 2.0 eV at different temperatures such as 300 K, 700 K, and 1200 K and are demonstrated in Fig. 10(c) and (d).The value of s/s is found to be zero in the chemical range from 0.0 eV to 1.0 eV, aer which it starts to increase with both sides of chemical potential and is observed to have opposite behavior above ±1.5 eV for LiZrCoGe in the studied chemical potential range.A similar trend is followed for LiZrCoSn and was observed to have opposite behavior of increasing with the temperature above ±1.3eV.The maximum value of s/s was found to be at the negative side of the chemical potential m-3 F , i.e., for p-type in both compounds.The variation of electronic thermal conductivity (k e /s) with the chemical potential is also studied and presented in Fig. 10(e) and (f).k e /s is found to be zero in the chemical potential range from 0.3 eV to 0.7 eV.The electronic thermal conductivity of both compounds is found to be increasing with temperature as we go from 300 K to 1200 K.The chemical potential dependence of the power factor (PF) has been studied for both compounds in the chemical potential range from −2.0 eV to 2.0 eV at 300 K, 700 K, and 1200 K, respectively, and is illustrated in Fig. 10(g) and (h).LiZrCoGe possesses a peak value of power factor of 2.2 × 10 12 W K −2 m −1 s −1 at −0.2 eV whereas LiZrCoSn has an optimal value of 1.8 × 10 12 W K −2 m −1 s −1 at −0.2 eV.The variation of gure of merit (ZT e ) with the chemical potential has been studied in the chemical potential range of −2.0 eV to 2.0 eV and is presented in Fig. 10(i) and (j) at different temperatures of 300 K, 700 K, and 1200 K, respectively.It can be seen from Fig. 10(i) that both compounds attain ZT e = 0.8 at room temperature in the vicinity of the Fermi level and reaches an optimal value of approximately unity at 0.3 eV.In the p-type region, ZT e increases with the temperature whereas in the range of 0 eV to 1.3 eV in the n-type region, ZT e decreases with the temperature in the case of LiZrCoGe and LiZrCoSn.

Conclusion
With the aid of DFT as implemented in the WIEN2k code, the structural, magnetic, electrical, elastic, vibrational, optical, thermodynamic, and thermoelectric properties of quaternary LiZrCoX (X = Ge, Sn) compounds are thoroughly examined.The computed equilibrium structural data are in good agreement with the prevailing similar theoretical and experimental literature.The studied compounds are found to be stable in Type-3 crystal structure as seen from the volume optimization curves.The tabulated cohesive energy, formation energy, and phonon band spectrum of these compounds depict their chemical, thermodynamic, as well as dynamic stability in the cubic phase.The computed second-order elastic constants of LiZrCoX compounds satisfy the mechanical stability of these compounds.The variation in the optical properties such as dielectric function, refractive index, extinction coefficient, absorption coefficient, reectivity, energy loss function, and optical conductivity with the incident electromagnetic radiations up to 12 eV are also studied.As an outcome, LiZrCoX compounds have the potential to be novel and intriguing thermoelectric materials that can also be used in optoelectronic and energy harvesting systems.

Fig. 4
Fig. 4 Band structure and density of states plot for (a) LiZrCoGe and (b) LiZrCoSn.
Fig. 5 Partial density of states for (a) LiZrCoGe and (b) LiZrCoSn.

Fig. 7
Fig. 7 Variation of (a) volume V, (b) bulk modulus, (c) specific heat at constant volume C v , (d) specific heat at constant pressure C p , (e) entropy S, and (f) linear expansion with pressure and temperature for LiZrCoGe.

Fig. 8
Fig. 8 Variation of (a) volume V, (b) bulk modulus, (c) specific heat at constant volume C v , (d) specific heat at constant pressure C p , (e) entropy S, and (f) linear expansion with pressure and temperature for LiZrCoSn.